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The Lattice of L-Group Varieties

Smith, Jo E.
http://rave.ohiolink.edu/etdc/view?acc_num=bgsu1566302070903521

Abstract Details

1976, Doctor of Philosophy (Ph.D.), Bowling Green State University, Mathematics.
For any type of abstract algebra, a variety is an equationally defined class of such algebras. Varieties were first introduced in the 30's and since then have been used successfully as a tool in the study of group theory. More recently, varieties of lattice-ordered groups (l-groups) have been found to be of interest, with various results obtained concerning particular varieties of l-groups. A more comprehensive study of l-group varieties was made by Martinez, in which the set of all such varieties was shown to form a lattice L under set inclusion with a compatible associative multiplication operation. The purpose of this dissertation is to continue the study of this lattice. Certain of the Scrimger varieties Sn (n and m) had been shown to be minimal non-abelian varieties n in L, In this work, it is shown that these varieties can be used to produce varieties minimal with respect to properly containing various other varieties in L. Also discussed are the relations among the Sn's, and it is established that all infinite collections of Sn's have the same least upper bound in L. Martinez has also classified l-groups using torsion classes, a generalization of the idea of varieties. It is proved here that L is not a sublattice of the lattice of torsion classes, and conditions are given for when joins in the two lattices will differ. Finally, an inductive construction is given for nested families of l-groups and corresponding varieties, which generalize the Scrimger varieties. These new varieties are examined in some detail and used to prove that in general Sn = Ln (the variety of l-groups for which nth powers commute).
W. Charles Holland (Advisor)
Mathematics

Recommended Citations

Citations

  • Smith, J. E. (1976). The Lattice of L-Group Varieties [Doctoral dissertation, Bowling Green State University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=bgsu1566302070903521

    APA Style (7th edition)

  • Smith, Jo. The Lattice of L-Group Varieties. 1976. Bowling Green State University, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=bgsu1566302070903521.

    MLA Style (8th edition)

  • Smith, Jo. "The Lattice of L-Group Varieties." Doctoral dissertation, Bowling Green State University, 1976. http://rave.ohiolink.edu/etdc/view?acc_num=bgsu1566302070903521

    Chicago Manual of Style (17th edition)

bgsu1566302070903521
204
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